Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Necessary and sufficient condition for the Lamperti invariance principle

Authors: Alfredas Rackauskas and Charles Suquet
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal: Theor. Probability and Math. Statist. 68 (2004), 127-137
MSC (2000): Primary 60F17; Secondary 60B12
Published electronically: May 24, 2004
MathSciNet review: 2000642
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of random variables with null expectation and variance one, $S_n:= X_1+\dots + X_n$, and $\xi_n$ the random polygonal line with vertices $(k/n,S_k)$, $k=0,1,\dots,n$. By a theorem of Lamperti (1962), if $X_1$ has a moment of order $p>2$, then $n^{-1/2}\xi_n$ weakly converges to the standard Brownian motion $W$ in the Hölder space $H_\alpha$ for $\alpha<1/2-1/p$. We prove that a necessary and sufficient condition for the $H_\alpha$-weak convergence of this process to $W$ is that $\mathsf{P}(\vert X_1\vert>t)=o(t^{-p(\alpha)})$, where $p(\alpha)=1/(1/2-\alpha)$. As an illustration, we present an application to the change point detection under the epidemic alternative.

References [Enhancements On Off] (What's this?)

  • 1. Z. Ciesielski, On the isomorphisms of the spaces $H_\alpha$ and $m$, Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8 (1960), 217-222. MR 24:A2234
  • 2. M. Csörgo and L. Horváth, Limit Theorems in Change-Point Analysis, John Wiley & Sons, New York, 1997.
  • 3. R. V. Erickson, Lipschitz smoothness and convergence with applications to the central limit theorem for summation processes, Ann. Probab. 9 (1981), 831-851. MR 83h:60006
  • 4. D. Hamadouche, Invariance Principles in Hölder Spaces, Portugaliae Mathematica 57 (2000), 127-151. MR 2001g:60075
  • 5. B. S. Kashin and A. A. Saakyan, Orthogonal Series, Translations of Mathematical Monographs, vol. 75, AMS, Providence, Rhode Island, 1989. MR 90g:42001
  • 6. G. Kerkyacharian and B. Roynette, Une démonstration simple des théorèmes de Kolmogorov, Donsker et Itô-Nisio, C. R. Acad. Sci. Paris Série I 312 (1991), 877-882. MR 92g:60009
  • 7. J. Lamperti, On convergence of stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 430-435. MR 26:804
  • 8. A. Rackauskas and Ch. Suquet, Central limit theorem in Hölder spaces, Probab. Math. Stat. 19 (1999), 155-174. MR 2000k:60012
  • 9. -, On the Hölderian functional central limit theorem for i.i.d. random elements in Banach space, Proceedings of the Fourth Hungarian Colloquium on Limit Theorems of Probability and Statistics (1999), Pub. IRMA Lille 50-III, 1999. MR 2004a:60004
  • 10. L. A. Shepp, Radon-Nikodym derivatives of Gaussian measures, Annals of Mathematical Statistics 37 (1966), 321-354. MR 32:8408

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60F17, 60B12

Retrieve articles in all journals with MSC (2000): 60F17, 60B12

Additional Information

Alfredas Rackauskas
Affiliation: Vilnius University, Department of Mathematics, Naugarduko 24, Lt-2006 Vilnius, Lithuania

Charles Suquet
Affiliation: Université des Sciences et Technologies de Lille, Laboratoire de Mathématiques Appliquées F.R.E. CNRS 2222, Bât. M2, U.F.R. de Mathématiques, F-59655 Villeneuve d’Ascq Cedex France

Received by editor(s): April 4, 2002
Published electronically: May 24, 2004
Additional Notes: Research supported by a cooperation agreement CNRS/LITHUANIA (4714).
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society